Antarctic Whale Project: Single Species
Ben Weinstein
2016-10-18 09:52:36
0.1 By Month
0.2 CCAMLR Units
0.3 Distance
0.4 Time
0.5 Velocity
0.6 Angles
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
How did the filter change the extent of tracks?
Look at the observations were defined into tracks.
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])
#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha_mu[state[i,g,t-1]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha_mu[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(20, 20)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
gamma[2] ~ dbeta(1,5) ## gamma for state 2
dev ~ dunif(0.1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[1] <- gamma[2] + dev
##Behavioral States
#Hierarchical structure across motnhs
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)
#latitidunal argos error
argos_alpha~dunif(0,10)
#correlation in argos error
rho ~ dunif(-1, 1)
}"
,fill=TRUE)sink()
## user system elapsed
## 1.410 0.056 893.4381.4 Chains
## Type Size PrettySize Rows Columns
## jagM rjags.parallel 75155096 [1] "71.7 Mb" 6 NA
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## m ggmap 13116480 [1] "12.5 Mb" 1280 1280
## b SpatialPointsDataFrame 4875992 [1] "4.7 Mb" 3354 47
## fccamlr data.frame 1649608 [1] "1.6 Mb" 41160 7
## sxy list 1184688 [1] "1.1 Mb" 32 NA
## mxy grouped_df 954216 [1] "931.9 Kb" 2924 52
## d SpatialPointsDataFrame 939008 [1] "917 Kb" 3354 47
## oxy data.frame 882256 [1] "861.6 Kb" 3354 47
## data list 724544 [1] "707.6 Kb" 9 NA## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1552602 83.0 2637877 140.9 2637877 140.9
## Vcells 15509395 118.4 29035403 221.6 35733381 272.7
Look at the convergence of phi, just for an example
Same section of chain for state
Overall relationship between phi and state, nice test of convergence.
1.4.1 Compare to priors
1.5 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1] -0.42707156 -2.937763398 1.5979076
## 2 alpha_mu alpha_mu[2] 2.04043806 0.432851690 3.6966692
## 3 gamma gamma[1] 0.41540786 0.281960880 0.5569139
## 4 gamma gamma[2] 0.09060005 0.005951651 0.2378229
## 5 theta theta[1] -0.06096017 -0.257053471 0.1281514
## 6 theta theta[2] 2.75650002 1.773922586 3.8819279
2 Behavioral Prediction
Relationship between phi and state
2.1 Spatial Prediction
2.2 Confidence in state estimate
Red is traveling, blue is foraging, midpoint is 0
2.3 By individual
2.4 Compared to CMLRR regions
2.5 Autocorrelation in behavior
2.6 Location of Behavior
3 Overlap with Krill Fishery
4 Time spent in grid cell
4.1 Traveling
## Type Size PrettySize Rows Columns
## pc tbl_df 74459632 [1] "71 Mb" 1430000 10
## a tbl_df 19846152 [1] "18.9 Mb" 496000 7
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## temp ggmap 13116816 [1] "12.5 Mb" 1280 1280
## b SpatialPointsDataFrame 4875992 [1] "4.7 Mb" 3354 47
## fccamlr data.frame 1649608 [1] "1.6 Mb" 41160 7
## d SpatialPointsDataFrame 939008 [1] "917 Kb" 3354 47
## mxy data.frame 929032 [1] "907.3 Kb" 2748 58
## oxy data.frame 882256 [1] "861.6 Kb" 3354 47
## traveling data.frame 833808 [1] "814.3 Kb" 2461 58## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1615418 86.3 2637877 140.9 2637877 140.9
## Vcells 18313403 139.8 41986979 320.4 41979005 320.3